IVP: A function f has the intermediate value property on an interval [a,b] if for all x<y in [a,b] and all L between f(x) and f(y), it is always possible to find a point c in (x,y) where f(c)=L And IVT: If f:[a,b]->R is continuous, and if L is a real number satisfying f(a)<L<f(b) or f(a)>L>f(b) then there exists a point c in (a,b) where f(c)=L Both definitions seem very similar to me, what purpose do they serve individually?

Kade Rosales

Kade Rosales

Answered question

2022-08-11

IVP: A function f has the intermediate value property on an interval [ a , b ] if for all x < y in [ a , b ] and all L between f ( x ) and f ( y ), it is always possible to find a point c ( x , y ) where f ( c ) = L

And

IVT: If f : [ a , b ] R is continuous, and if L is a real number satisfying f ( a ) < L < f ( b ) or f ( a ) > L > f ( b ) then there exists a point c ( a , b ) where f ( c ) = LBoth definitions seem very similar to me, what purpose do they serve individually?

Answer & Explanation

Gemma Conley

Gemma Conley

Beginner2022-08-12Added 11 answers

One way to state the IVT is: If f : [ a , b ] R is continuous, then f has the IVP.
So the IVP is a property, the IVT is the statement that continuous functions have the IVP!

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